Improved bounds for radial projections in the plane

TL;DR

This paper establishes improved lower bounds for the Hausdorff dimension of radial projections of sets in the plane, advancing understanding of geometric measure theory related to projections.

Contribution

It provides a new lower bound for the Hausdorff dimension of radial projections of sets in the plane, generalizing previous results.

Findings

New lower bound for radial projections in the plane

Applicable to Borel sets with positive Hausdorff dimension

Advances in geometric measure theory

Abstract

We improve the best known lower bound for the dimension of radial projections of sets in the plane. We show that if $X,Y$ are Borel sets in $\R^2$, $X$ is not contained in any line and $\dim_H(X)>0$, then $$\sup\limits_{x\in X} \dim_H(\pi_x Y) \geq \min\left\{(\dim_H(Y) + \dim_H(X))/2, \dim_H(Y), 1\right\},$$ where $\pi_x Y$ is the radial projection of the set $Y$ from the point $x$.